Quicksort and selection

1. Quicksort and selection Prof. Ramin Zabih http://cs100r.cs.cornell.edu

2. Administrivia • Assignment 2 is out, due in 2 pieces • Small amount is due this Friday • Most of it is due next Friday • Quiz 1 and Assignment 1 are graded • You can check CMS • Optional evening lecture(s) coming soon! • “Kindergarten cop” will be in a month or so • Quiz 2 on Tuesday 9/18 • Coverage through the next lecture

3. Recap from last time • Big-O notation allows us to reason about speed without worrying about • Getting lucky on the input • Depending on our hardware • Repeatedly removing the biggest element allows us to find the kth largest • Problem is called “selection” • Algorithm is quadratic, O(n2)

4. How to do selection better? • If our input were sorted, we can do better • Given 100 numbers in increasing order, we can easily figure out the 5th biggest or smallest • Very important principle! • Divide your problem into pieces • One person (or group) can provide sort • The other person can use sort • As long as both agree on what sort does, they can work independently • Can even “upgrade” to a faster sort • We will do this in CS100R

5. How to sort? • Sorting is an ancient problem, by the standards of CS • First important “computer” sort used for 1890 census, by Hollerith • Strangely enough, we will talk about the census again shortly (the 1900 census) • There are many algorithms • We will focus on quicksort

6. Quicksort summary • Pick an element (pivot) • Partition the array into elements <= pivot and > pivot • Quicksort these smaller arrays separately • Example: [10 2 5 30 4 8 19] • If the pivot is 8 we need to sort: [2 5 4 8] and [10 30 19] • Sort these using quicksort

7. The selection problem • Rev. Charles L. Dodgson’s problem • Based on how to run a tennis tournament • Specifically, how to award 2nd prize fairly

8. Quicksort and the pivot • There are lots of ways to make quicksort fast, for example by swapping elements • We will cover these in section • Notice that with a bad choice of pivot this algorithm does quite poorly • Suppose we happen to always pick the largest element of the array, or the smallest? • What does this remind you of? • When can the bad case easily happen?

9. Modifying quicksort to select • Suppose that we want to find the 5th largest element (4 are larger) • Quicksort kind of helps us do this already • When we partition the array, we know the element we want is in one smaller array! • So, we can figure out which smaller array we need to search • For example, if we have 10 elements, 7 <= pivot and 3 > pivot • We need to search for the 2nd largest element in the smaller array with 7 elements

10. Modified quicksort example • Example: find 5th largest (< 4) in: [8 3 4 6 3 8 5 10 10 7] • The ones bigger than 8 are: [ 10 10 ] • So we need the 3rd largest (< 2) in: [8 3 4 6 3 8 5 7]

11. Find element < k others in A MODIFIED QUICKSORT: • Pick an element in A as the pivot, call it x • Divide A into A1 (<x), A2 (=x), A3 (>x) • If k < length(A3) • Find the element < k others in A3 • If k > length(A2) + length(A3) • Find the correct element in A1 • Let j = k – [length(A2) + length(A3)] • Find the element < j others in A1 • Otherwise, return x

12. What is the complexity of: • Finding the 1st element? • O(1) (effort doesn’t depend on input) • Finding the biggest element? • O(n) (constant work per input element) • Finding the median, by repeatedly finding and removing the biggest element? • O(n2) (linear work per input element) • Finding the median using modified quicksort? • O(n2) (linear work per input element)

13.  What is the complexity of: • Finding the 2nd biggest element (>all but 1)? The 3rd biggest (> all but 2)? • What do you think? • It’s actually O(n) • We do 2 (or 3) “find biggest” operations • Each of which takes O(n) time • Finding the element bigger than all but 5%? • Assume we do this by repeated “find biggest” • What if we use modified quicksort?